Some formulae relating modular representations of elementary abelian p-groups

Abstract Let $$p>0$$ p > 0 be a prime, $$\Bbbk $$ k a field of characteristic p and G an elementary abelian p -group of order $$q=p^n$$ q = p n . Let W be an indecomposable $$\Bbbk G$$ k G -module of dimension 2 and define $$V_i = S^{i-1}(W)^*$$ V i = S i - 1 ( W ) ∗ for each $$i=1, \ldots $$ i = 1 , … , q. We show that $$V_2 \otimes V_i \cong V_{i+1} \oplus V_{i-1}$$ V 2 ⊗ V i ≅ V i + 1 ⊕ V i - 1 provided i is not divisible by p . Our results generalise results of Almkvist and Fossum (1978) for representations of cyclic groups of order p . We show how our results give formulae for the direct sum decomposition of $$V_i \otimes V_j$$ V i ⊗ V j for all $$i i < p and $$j j < q modulo summands projective to $$M:= \bigoplus _{r=0}^{p^{n-1}}V_{rp}$$ M : = ⨁ r = 0 p